SCHUBERT CLASSES OF A LOOP GROUP

SCHUBERT CLASSES OF A LOOP GROUP

PETER MAGYAR

In these notes, we survey the homology of the loop group ΩK of a compactgroup K, also known as the affine Grassmannian Gr of a complex loop groupG[C((t))] . Using the Bott picture [B] of H∗(ΩK), the homology algebra orPontryagin ring, we obtain two new results:

A. Factorization of affine Schubert homology classes.

B. Definition of affine Schubert polynomials representing the affine Schu-bert homology classes in all types, in terms similar to ordinary Schu-bert polynomials (Demazure operators on the Borel picture of thecohomology of the flag variety H∗(K/ T )).

The philosophy is that working in the topological category (with compactrather than algebraic groups) gives extra flexibility which simplifies the con-structions. For example, homology factorization happens because the Bott-Samelson variety is topologically a direct product, even though algebraicallyit is a non-trivial fiber bundle. Bott and Samelson exploit this flexibility toconstruct the Pontryagin ring H∗(ΩK) as the symmetric algebra of the ho-mology H∗(K/ T ). Once this construction is made, the computation of theSchubert classes becomes a standard exercise in the theory of Schubert poly-nomials in the equivariant cohomology H∗

T (K/ T ). Thus, one may think ofsingular homology as a flabby version of equivariant K-theory and represen-tation theory (Demazure modules). We work with rational (co)homologythroughout this note, but with some care over denominators the resultscould be refined to integer coefficients.

In this note, for concreteness we work with G = SLnC, but unless oth-erwise noted everything generalizes to a simple, simply connected linearalgebraic group of arbitrary type. In §1, we survey the relevant background.We give self-contained statements of Theorem A in §2.5 and Theorem B forG = SLn in §3.6. In an appendix §4, we discuss weak order and factorizationin the affine Weyl group.

Our formulas are “global,” analogous to the Schubert polynomials inH∗(G/B) and H∗

T (G/B). Let us also mention the unpublished work ofDale Peterson [P], which contains a different and very powerful point ofview on these topics based on the local picture of equivariant cohomology atT -fixed points (the nil-Hecke ring of Kostant-Kumar). Peterson shows that

Date: September 2005. Revised March 2007.

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2 MAGYAR

the structure coefficients for H∗(ΩK) are certain Gromov-Witten invariantsfor the flag variety G/B, structure coefficients of the quantum cohomologyring qH∗(G/B). An alternative proof of Peterson’s theorem might well bepossible by combining Bott’s realization of H∗(ΩK) with the known struc-ture of qH∗(G/B).

Using Peterson’s picture, Thomas Lam [L] has shown that one versionof Lapointe & Morse’s restricted Schur polynomials represent the Schubertclasses of H∗(ΩK), so that these polynomials are identical to our Schubertpolynomials for G = SLn . It should be possible to extend this theoremusing the computation of affine Schubert polynomials in Theorem B.

We wish to thank Mark Shimozono, who suggested the questions we ex-amine here.

1. Background on topology and combinatorics

1.1. Loop group and affine Grassmannian. Let G((t)) := SLnC((t)) bethe group of determinant-one matrices with entries in the field of formalLaurent series C((t)). Then G((t)) naturally includes the analytic functionsf : C× → G, by identifying a function with its Laurent series. Let G[[t]] :=SLnC[[t]] be the subgroup of determinant-one matrices with entries in theformal Taylor series C[[t]]. We may regard G[[t]] as a maximal parabolicsubgroup of G((t)). The affine Grassmannian is the quotient space Gr :=G((t))/ G[[t]] , which we can also realize as the C[[t]]-lattices with a fixedvirtual dimension in the C((t))-vector space C((t))n.

Let K be a maximal compact subgroup of G , namely K = SUn =A ∈ Mn(C) | AAt = I, the group of matrices whose column vectorsform an oriented orthonomal basis for Cn under the Hermitian inner prod-uct 〈(a1, . . . , an), (b1, . . . , bn)〉 :=

∑i aibi. Recall that K ⊂ G is a maximal

compact subgroup, a real Lie group which is not complex algebraic.Let LK denote the group of real analytic loops1 in K, the maps f : S1 →

K, with multiplication performed pointwise: (f · g)(t) := f(t) · g(t) ∈ K.Considering t ∈ S1 ⊂ C× as a complex parameter, any analytic loop f(t) ∈LK has a Laurent series, and we can consider LK ⊂ G((t)).

Let ΩK := f(t) ∈ LK | f(1) = 1K be the normal subgroup of basepointpreserving loops. We can realize the affine Grassmannian as: Gr ∼= ΩK .This holds because Gram-Schmidt orthogonalization allows us to write anyf(t) ∈ G((t)) uniquely as f(t) = fK(t)·fP (t), where fK(t) ∈ ΩK and fP (t) ∈G[[t]], so G((t)) = ΩK ·G[[t]], a product of transverse subgroups. Thus thesemi-infinite group ΩK provides a canonical slice (set of representatives)for the doubly infinite quotient Gr = G((t))/G[[t]]. This is what makes thetopological theory (using compact groups) simpler than the algebraic theory(using algebraic groups).

1More precisely, we mean the maps which extend to meromorphic functions f : D → Kon the closed unit disk D ⊂ C, with poles only allowed at t = 0.

SCHUBERT CLASSES OF A LOOP GROUP 3

It will sometimes be convenient to identify a basepoint-preserving loop inΩK with the class of its translates by K. This leads to the realization ofthe affine Grassmannian as:

Gr ∼= LK / (LK ∩G[[t]]) = LK/K ∼= ΩK .

where K ⊂ LK is the group of constant loops f(t) = k ∈ K. Indeed, we canwrite any loop in LK as: f(t) = f(t)f(1)−1 · f(1) ∈ ΩK ·K, and this givesa natural homeomorphism LK/K ∼= ΩK. (This is not a group morphism,however.)

1.2. Pontryagin ring. Let us consider the singular homology of Gr = ΩK.Although ΩK is infinite-dimensional, it is tamely infinite, meaning we canfind an increasing union of finite-dimensional compact Hausdorff subspacesΩ1 ⊂ Ω2 ⊂ · · · ⊂ ΩK with ΩK =

⋃j Ωj . (We can choose Ωj from among the

Schubert subvarieties we will define below.) Thus H∗(ΩK) = lim→

H∗(Ωj),

a direct limit.2

We have seen that Gr = ΩK is topologically a loop group with a con-tinuous (not complex algebraic) multiplication mult : ΩK × ΩK → ΩK.This induces on the singular homology H∗(ΩK) := H∗(ΩK, Q) a multiplica-tion mult∗ : H∗(ΩK)⊗H∗(ΩK)→ H∗(ΩK), which together with the usualaddition defines the so-called Pontryagin ring.

It is a basic fact that the Pontryagin multiplication is commutative. Fur-thermore, let us consider a different multiplication conc : ΩK × ΩK → ΩKdefined by concatenating loops:

(f · g)(e2πix) :=

f(e4πix) , 0 ≤ x ≤ 12

g(e4πix) , 12 ≤ x ≤ 1 .

Then the new induced multiplication conc∗ on H∗(ΩK) is the same as mult∗ .

Proof: We construct a homotopy of loops f · g ≈ f · g ≈ g · f . For each−1 ≤ ε ≤ 1, let pε : [0, 1]→ [0, 1]× [0, 1] be a path in the unit square startingat (0, 0) and ending at (1, 1), such that p−1 goes along the left and top bound-aries, p0 goes along the diagonal, and p1 goes along the bottom and rightboundaries. Define H : [0, 1]×[0, 1]→ K by H(x1, x2) := f(e2πix1) g(e2πix2).Then defining hε : S1 → K by hε(e2πix) := H(pε(x)) gives a continuous fam-ily of loops with h−1 = f · g, h0 = f · g, and h1 = g · f .

As a corollary, the fundamental group of K (or of any Lie group) isabelian, since multiplying in π1(K) means concatenating loops. Of course, inour case G = SLnC and K = SUn are simply connected, so the fundamentalgroup is trivial (and the loop group ΩK is connected). On the other hand,

2Indeed, we can thicken each Ωj into a normal neighborhood Uj ⊂ ΩK which is re-tractable to Ωj . Since a singular cycle has compact support, it must live in a sufficientlylarge Uj , so that it is homologous to a cycle in Ωj .

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if we take the quotient of K by the center

Z = Z(G) = Z(K) =

e2πin

j I | j = 0, 1, . . . , n−1

,

we get the adjoint groups Gad = G/Z and Kad = K/Z. Since G → Gad

and K → Kad are universal covering maps with fiber Z, we have π1(Gad) ∼=π1(Kad) ∼= Z ∼= Z/nZ, a cyclic group of order n.

1.3. Coweight lattice and adjoint group. The subgroup of diagonalmatrices T ⊂ K is a maximal torus: T ∼= (S1)n−1. Letting Rn = Re1⊕· · ·⊕Ren, we may realize the Lie algebra t = Lie(T ) as:

t ∼= (x1, . . . , xn) ∈ Rn | x1 + · · ·+ xn = 0 .

The exponential map is a group morphism and universal covering map:exp : t→ T , (x1, . . . , xn) 7→ diag(e2πix1 , . . . , e2πixn). Its kernel is the corootlattice:

Q∨ := Ker(exp : t→ T )

= λ ∈ Zn | λ1+ · · ·+λn = 0 =n−1⊕i=0

Zα∨i ,

where α∨i := ei − ei+1 are the simple coroots. If we lift the center Z =Z(G) = Z(K) ⊂ T up to t, we get the coweight lattice:

P∨ := exp−1(Z) = λ | λ ∈ Zn =n−1⊕i=1

Z$∨i ,

where we use the projection − : Rn → t , λ := λ − λ1+···+λnn (e1+ · · ·+en) ,

and $i := e1+ · · ·+ ei ∈ 1nZn are the fundamental coweights. Exponentiat-

ing P∨ back into T , we see that Z ∼= P∨/ Q∨.Let us identify each vector λ ∈ Q∨ with the straight-line path [0, 1] → t,

x 7→ xλ and then exponentiate both sides. This makes λ ∈ Q∨ corre-spond to the loop S1 → T , t 7→ tλ := diag(tλ1 , . . . , tλn), and in factQ∨ ∼= Hom(S1, T ), the homomorphisms from the circle group to the torus.Similarly, P∨ ∼= Hom(S1, T ad), where T ad := T/Z is the maximal torus ofKad = K/Z.

Consider ΩKad, the loops in the adjoint group. The connected compo-nents of ΩKad are by definition the elements of the fundamental groupπ1(Kad). Also, the identity component ΩKad

(consisting of homotopi-cally trivial loops) is normal, and the quotient is the component groupΩKad / ΩKad

∼= π1(Kad).

We can understand ΩKad purely in terms of K. Every loop h : [0, 1] →Kad with h(0) = h(1) = 1 lifts uniquely to a path h : [0, 1]→ K starting ath(0) = 1 and ending at some h(1) = z ∈ Z. The map ΩKad → Z , h(x) 7→h(1) = z realizes the quotient map to the component group Z ∼= π1(Kad) .

A homotopically trivial loop h(x) ∈ ΩKad lifts to a loop in K, meaning

h(1) = 1 and h(x) = f(e2πix) for f(t) ∈ ΩK, and this identifies ΩKad∼= ΩK.


As for a component corresponding to z = exp(µ) ∈ Z with µ ∈ P∨, themap ΩK → ΩKad, h(x) 7→ h(x) · exp(2πixµ) (pointwise product) gives ahomeomorphism from ΩK to the z-component of ΩKad.

1.4. Affine Weyl group. Let W = Sn be the finite Weyl group, whoseelements w ∈W act on t ⊂ Rn by permuting the n basis elements e1, . . . , en,and on Rn→→ t∗ by permuting the dual basis elements x1, . . . , xn . Then Wis a Coxeter group whose reflections correspond to the roots of G: for eachpair of a coroot α∨ = ei−ej ∈ t and its corresponding root α = xi−xj ∈ t∗,we have the reflection rα : t→ t, µ 7→ µ− 〈α, µ〉α∨. The simple reflectionsare s1, . . . , sn−1 for αi = xi − xi+1.

We also have the affine Weyl group W = W ×Q∨, a semi-direct productin which W acts on Q∨. For w ∈ W , λ ∈ Q∨, an element wtλ ∈ W actson t ⊂ Rn by the affine transformation µ 7→ w(µ + λ), and wtλ = tw(λ)w.Then W is a Coxeter group whose (affine) reflections correspond to affineroots, pairs β = (α, k) with α a root of W and k ∈ Z: namely, rbβ = r(α,k) :=

rαtkα∨ = t−kα∨rα.The affine simple roots are α1 = (α1, 0), . . . , αn−1 = (αn−1, 0), and

α0 := (−θ, 1), where θ = x1−xn is the highest root and θ∨ = e1−en itscoroot.3 The affine simple reflections are si := rbαi

for i = 0, . . . , n−1. Notethat s1, . . . , sn−1 ∈W ⊂ W , and s0 = rθ t−θ∨ = tθ

∨rθ.

We may consider the Weyl group as the quotient W ∼= NK(T ) / T , wherethe normalizer NK(T ) consists of scaled permutation matrices. We willdenote a representative of w ∈ W by

w ∈ NK(T ). Similarly, we have:

W ∼= NLK(L0T ) /L0T , where L0T is the group of loops f : S1 → T whichextend to analytic functions on the closed unit disk D ⊂ C. Furthermore,the loop tλ ∈ Hom(S1, T ) ⊂ NLK(L0T ) gives a canonical representative forthe corresponding element tλ ∈ W , which justifies using the same symbolfor both.

There is a technical unpleasantness in this definition of W , however. Theadjoint action of NLK(L0T ) on t has kernel LT , not just L0T , so the adjointaction does not produce the affine action of W on t. This can be remediedby passing to the central extension of LK (or of G((t))), which is the trueaffine Kac-Moody group.

1.5. Extended affine Weyl group. We also have the extended affine Weylgroup W = W × P∨ consisting of affine transformations wtλ with w ∈ W

and λ ∈ P∨, so that W/W ∼= P∨/ Q∨ ∼= Z. There is a subgroup Σ ⊂ W

such that W = Σ · W , a semi-direct product in which the subgroup Σ ∼= Z

acts on W . In fact, Σ = StabfW (A0), where A0 is the fundamental alcove, asimplex which is a fundamental domain of W acting on t:

A0 := λ ∈ t | 〈αi, λ〉 ≥ 0, 〈θ, λ〉 ≤ 1 .

3Warning: For G not simply laced, θ∨ is not the highest root of the dual root system.

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We can explicitly describe the isomorphism Σ ∼= P∨/ Q∨ as follows:

Σ ←→ P∨/ Q∨

σ 7−→ η = σ(0)σ = tηw0wη ←−a λ ,

where η is the minimal dominant coweight with η = λ mod Q∨; and w0, wη ∈W are respectively the longest element and the longest element with w(η) =η. The η which occur give canonical representatives for P∨/ Q∨ : they areprecisely the minuscule4 fundamental coweights η = $∨

i (and η = 0 forσ = id). We can lift σ to LK as:

σ := tη

w0

wη ∈ LK.

Every v ∈ W has a unique decompostion v = σvv with σv ∈ Σ andv ∈ W . To extend the length function to W we let `(v) := `(v), so thatΣ = v ∈ W | `(v) = 0.

1.6. Geometry of the center. We record some irrelevant facts about thegeometry of the center Z and the adjoint group Kad = K/Z.

Consider Kad acting on K by conjugation. This clearly commutes withmultiplication by a central element z ∈ Z given by: τ : K → K, k 7→z · k. (This is a deck-shuffling tranformation of the universal covering mapK → Kad.) Both of these actions are isometries of K (under a suitableRiemannian metric).

Since every conjugacy class of K intersects T in an orbit of the Weylgroup W , we can construct the quotient as:

K/Kad ∼= T/W ∼= t/(W ×Q∨) ∼= A0 ,

where A0 is the fundamental alcove of the previous section. Now, τ = τz

induces an isometry of the quotient K/Kad, namely:

τ : A0 → A0

p 7→ p + η mod (W ×Q∨) ,

where η = ηz is the minimal dominant coweight with exp(2πiη) = z : this isalways a minuscule fundamental coweight, or η = 0 if z = 1. The formulaσ = tηw0wη lifts τ to a map σ : t→ t with σ(A0) = A0.

The geometry of A0 is encoded in the affine Dynkin diagram Dyn: forexample, the walls of A0 correspond to the vertices of Dyn, the affine wallcorresponding to the distinguished 0-node. A non-trivial isometry σ = σz

of A0 defines a graph automorphism of Dyn which moves the 0-node to thenode i corresponding to the minuscule fundamental coweight $∨

i = η = σ(0).Thus we may embed Z → Sym(Dyn). Finally, Dyn is the Dynkin graph ofthe Kac-Moody group G((t)), and any element of Dyn induces an outerautomorphism of G((t)), LK, and ΩK.

4A coweight λ ∈ P∨ is minuscule if 〈α, λ〉 = 0 or 1 for all positive roots α.


To summarize the many guises of the center:

Z = Z(G) = Z(K)∼= P∨/ Q∨ = 0 ∪ η = $∨

i minuscule∼= W/ W = Σ = σ ∈W×P∨ | σ(A0) = A0∼= π1(Kad) ∼= ΩKad/ ΩK

→ Isometry(A0) ∼= Aut(Dyn) ∼= Out G((t)) .

1.7. Flag varieties. Let B ⊂ G, the Borel subgroup of upper-triangularmatrices in G. For a coweight λ ∈ P∨, we have the subgroup:

Gλ := StabG(tλ) := g ∈ G | gtλg−1 = tλ for all t ∈ C× ,

as well as the parabolic subgroup generated by these two subgroups: Pλ :=GλB ⊂ G. The compact version is Kλ := K ∩ P λ. Note that T ⊂ Kλ andKw(λ) = wKλw−1.

We will be concerned with $ := $∨1 = e1 ∈ t, the first fundamental

coweight, for which we have:

P$ = A = (aij) ∈ SLnC | ai1 = 0 for j = 2, . . . , n

and K$ = SUn ∩ (U1 × Un−1), where U1 × Un−1 denotes block-diagonaln×n unitary matrices. The corresponding partial flag variety is the complexprojective (n−1)-space:

X := G/P$ = K/K$ ∼= Pn−1 = Cn−0 / C× = S2n−1/S1 .

where we consider S2n−1 ⊂ R2n ∼= Cn . 5 Explicitly, we have the homeomor-phism:

X = G/P$ ∼→ Pn−1 = Cn−0 / C×

A = (aij) 7→ A · e1 = [a11 : · · · : an1] ,

so that a matrix goes to the projective point given by its first column.The full flag variety is the quotient space X = G/B, which can be realized

as the space of all nested sequences of complex subspaces (V1 ⊂ V2 ⊂ · · · ⊂Vn = Cn) with dim Vi = i. The compact description of this is: X ∼= K/(K ∩B) = K/T , which is the space of all orthonormal bases of Cn modulo scalingeach basis vector by S1 : i.e., the flag variety is the space of all splittings ofCn into orthogonal complex lines.

There is a right W -action on K/T : for x =xT and w =

wT , we let

x ·w :=xw T . This action is continuous but not holomorphic. For example,

if G = SL2C and w = (12) ∈ W , we havew =

»0 −11 0

–, and for a typical flag

5For n = 2, the circle bundle S1 → S3 → P1 ∼= S2 is known as the Hopf fibration.

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x =»

1 0a 1

–∈ G/B we compute:

G/B ∼= K/Tw→ K/T ∼= G/B

x mod B 7→ k mod T 7→ kw mod T 7→ y mod B[1 0a 1

]7→

[1/d −a/da/d 1/d

]7→

[−a/d −1/d

1/d −a/d

]7→

[1 0

−1/a 1

],

where d :=√

1 + aa. Here the first map is Gram-Schmidt orthogonaliza-tion, the second map is the W -action, and the third map is the holomorphiccoordinate function on a chart of K/T ∼= G/B ∼= P1. Thus, on this holo-morphic chart the map is w : a 7→ −1/a, which is anti-holomorphic (andorientation-reversing).

1.8. Classifying spaces. Consider the direct limit vector space C∞ :=limj→∞ Cj with the `2 metric and its unit sphere S∞, which is contractablesince its homology is trivial, and which has a free right action of the circlegroup S1. The basic example of a classifying space is the quotient B(S1) =P∞ := S∞/S1, the (tamely) infinite projective space.

Consider E := (S∞)nind, the space of linearly independent n-tuples of unit

vectors [~v1, . . . , ~vn] ∈ (C∞)n thought of as (∞×n)-matrices . This is con-tractable with a free right action of the torus (S1)n and of the unitary groupUn, so the quotients B(S1)n = E/(S1)n and BUn := E/Un are classifyingspaces of the respective groups. We have B(S1)n ∼= (P∞)n

ind, the space ofn-tuples of linearly independent lines in C∞, and BUn

∼= Gr(n, C∞), theGrassmannian of n-planes in C∞.

For a torus T ⊂ (S1)n, we define its classifying space as the free quotientBT := E/T . For any compact group with an n-dimensional representation,K ⊂ Un, we let BK := E/K. (The space E serves as a contractableprincipal bundle for any such T and K, so that E = ET = EK.)

In our case T ⊂ K = SUn , the space BT will be an S1-bundle overB(S1)n. Indeed, if we consider the dual tautological complex line bundleD := O(1, . . . , 1) over (P∞)n

ind , or simply over (P∞)n , we can constructour space BT as the associated S1-bundle (Thom bundle) of the complexline bundle D. An element of BT is essentially an n-tuple of complex lines(C~v1, . . . , C~vn) ∈ (P∞)n, together with a non-zero volume form6 on thecomplex n-space V = C~v1+ · · ·+ C~vn. We can also construct BK as theS1-bundle of the complex line bundle D = O(1) = ∧nTaut∗ → Gr(n, C∞) ∼=BUn : that is, an element of BK is a complex n-space V ⊂ C∞ endowedwith a non-zero volume form.

We have the T -equivariant cohomology ring H∗T (pt) := H∗(BT ) ∼= Sym(t∗),

the ring of polynomial functions on t ⊂ Rn. Indeed, for an element of thecoroot lattice λ ∈ Q∨ ⊂ t∗, we have the one-dimensional character exp(λ) :T → S1 and the associated complex line bundle Lλ := ET ×T Cexp(λ) →

6More precisely, a C-linear n-form on V which agrees in absolute value with the stan-dard form.


BT . Then λ is identified with the first Chern class c1(Lλ) ∈ H∗(BT ), thePoincare dual of the vanishing locus of a section of L. In our case we have:

H∗T (pt) = S := Q[y′1, . . . , y

′n]/(y′1 + · · ·+ y′n) ,

where y′1, . . . , y′n in (Rn)∗ forms a dual basis of e1, . . . , en ∈ Rn. For historical

and combinatorial reasons, we re-index the y-variables, letting:

yi := y′n+1−i = y′w0(i) .

We define a grading on Q[y] by writing: dimR(yi) = 2.The space BK has the same rational cohomology (but not the same

integer cohomology) as the finite quotient BT/W , where W permutes theentries of (~v1, . . . , ~vn) ∈ (P∞)n: that is, H∗(BK) ∼= H∗

T (pt)W ∼= SW , theW -invariant polynomials in S.

1.9. Cohomology of the flag variety. For a variety Y endowed with aT -action, we define the equivariant cohomology ring H∗

T (Y ) := H∗(YT ), theordinary cohomology of the tamely infinite-dimensional induced space:

YT := ETT× Y = (ET × Y ) / T ,

where the free T -action on ET×Y is given by: (e, y) · t = (et, t−1y) .Consider the flag variety X = K/T , with its T -equivariant cohomology

ring H∗T (X) := H∗(XT ). We have a homeomorphism:

XT = ETT× K/T

∼−→ BT ×BK

BT

([~v1, . . . , ~vn] , k) 7−→ ([~v1, . . . , ~vn] · k , [~v1, . . . , ~vn]) mod T 2 .

From this it is easy to see that H∗T (X) ∼= S ⊗SW S.

We can realize the ordinary cohomology H∗(X) as a quotient of theequivariant cohomology H∗

T (X) as follows. Choose an arbitrary basepointb ∈ BT , and consider the fiber bundle:

Xfib→ ET

T×X

proj→ BT ,

where fib : X∼→ X := proj−1(b), the fiber above the basepoint. Then

we have a surjective quotient map fib∗ : H∗T (X) → H∗(X) with kernel

(proj∗H∗T (pt)+), the ideal generated by the pullback classes of positive de-

gree. Writing this in terms of the polynomial algebra S, we get the Borelpicture of the cohomology:

H∗T (X) →→ H∗(X)

S ⊗SW

S →→ S/(SW+ ) .

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Furthermore, consider the projective space X = K/K$. The projectionπ : X 7→ X induces an inclusion of the equivariant cohomology:

π∗ : H∗T (X) → H∗

T (X)

S$ ⊗SW

S ⊂ S ⊗SW

S ,

where S$ denotes the invariants under W$, the Weyl group of K$. Simi-larly for the ordinary cohomology groups.

1.10. Cohomology generators and relations. We introduce coordinatesfor our cohomology rings. Recall that BT → (P∞)n possesses n dual tauto-logical bundles O(0, . . . , 1, . . . , 0). We let xi (resp. y′i) denote the Chern classof the ith bundle on the first factor (resp. the second factor) of BT ×BK BT ,and we recall our convention that yi := y′n+1−i . We now get:

H∗T (X) ∼= S ⊗

SWS ∼=

Q[x, y]J

:=Q[x1, . . . , xn, y1, . . . , yn](

x1 + · · ·+ xn = 0h(x) = h(y) ∀h ∈ SW

) ,

and:

H∗(X) ∼= S/(SW+ ) ∼=

Q[x]J0

:=Q[x1, . . . , xn]

(h(x) = 0 ∀h ∈ Q[x]W+ ).

We also have:

H∗T (X) ∼= S$ ⊗SW S ∼= Q[x, y]/J$ ,

where x := x1 and J$ := J ∩ S$, and similarly H∗(X) := Q[x]/J$0 . The

restriction maps fib∗ are realized by y = 0 , and π∗ is the obvious inclusion.We give Grobner bases for our ideals. (This is one of the few facts whose

generalization to arbitrary K is not known.) Consider the pure lexico-graphic term-order:

y1 < y2 < · · · < yn < x1 < x2 < · · · < xn .

Let hd(x[i]) denote the complete symmetric polynomial of degree d in thevariables x1, . . . , xi , and let ed(y) denote the elementary symmetric poly-nomial of degree d in the variables y1, . . . , yn. Then the unique reducedGrobner bases of our ideals are:

J =(h1(x[n]) , e1(y) ,

∑di=0 (−1)i hd−i(x[n+1−d]) ei(y) , d = 1, . . . , n

),

J0 =(h1(x[n]) , h2(x[n−1]) , . . . , hn(x[1])

),

J$ =(e1(y) , xn − xn−1e1(y) + · · ·+ (−1)n−1xen−1(y) + (−1)nen(y)

),

and J$0 = ( xn ) , where x = x1.


1.11. Poincare duality. It is possible (e.g., by the work of M.E. Kazarian)to define the Poincare duality isomorphism H2`

T (X) ∼→ HT∞−2`(X), where the

right side can be interpreted as the cycles of codimension 2` in the tamelyinfinite-dimensional space XT . Any T -invariant subvariety Y ⊂ X defines avariety YT ⊂ XT , and conversely any finite-codimension subvariety Y ′ ⊂ XT

defines a T -invariant subvariety:

Y ′⊥ = ⊥Y ′ := fib−1(Y ′ ∩ X) ⊂ X .

We have Y = (YT )⊥. Also codim(Y ⊂ X) = codim(YT ⊂ XT ), as well ascodim(Y ′⊂ XT ) = codim(Y ′

⊥⊂ X) provided the intersection Y ′ ∩ X istransversal.

The geometric operation Y ′ 7→ Y ′⊥ induces via Poincare duality the co-

homology map fib∗ : H∗T (X) → H∗(X) , where fib : X → XT is the above

fiber map: that is, fib∗[Y ′] = [Y ′⊥] . For a polynomial cohomology class

f(x, y) ∈ Q[x, y]/J ∼= H∗T (X) we have:

fib∗f = ⊥f(x, y) := f(x, 0) ∈ Q[x]/J0∼= H∗(X) .

The geometric operation Y 7→ YT does not induce a well-defined mapH∗(X) → H∗

T (X), since the equivariant class [YT ] depends not only uponthe homology class [Y ], but upon its T -action.

1.12. Demazure operations. For a root α ∈ t∗ , there are right and leftDemazure operations ∂R

α , ∂Lα defined on the equivariant cohomology of the

flag variety H∗T (X).

We first define the more elementary right operation ∂Rα acting on the or-

dinary cohomology H∗(X). We have the P1-bundle πα : K/T 7→ K/Kα,inducing on the cohomology H∗(K/T ) the pullback map (πα)∗ and thepush-forward map 7 (πα)∗ . Then we let: ∂R

α := (πα)∗(πα)∗ : H2`(X) →H2`−2(X) .

We can interpret ∂Rα geometrically by applying Poincare duality to identify

H2`(X) ∼= H2L−2`(X), where 2L = dimR(X). Namely, given a subvarietyY ⊂ K/T = X, we lift it to Y T ⊂ K, right multiply by Kα ⊃ T , and pushback down to get the variety DR

α (Y ) := Y Kα/T ⊂ X. Then we can expressthe homology operation in terms of fundamental classes as:

∂Rα : H2L−2`(X) −→ H2L−2`+2(X)

[Y ] 7−→

[DRα Y ] if dimR(DR

α Y ) = dimR(Y ) + 20 otherwise .

To define ∂Rα on H∗

T (X) = H∗(XT ), we can repeat the above word forword, replacing X = K/T by XT = KT/T . The equivariant and ordinaryDemazure operations, both denoted ∂R

α , commute with the fiber map ⊥ :H∗

T (X)→→H∗(X): that is, ⊥ ∂Rα = ∂R

α ⊥ .

7The so-called Gysin map, realized on DeRham cohomology as integrating over fibers.

12 MAGYAR

In terms of the coordinates H∗T (X) ∼= Q[x, y]/J , the operator ∂R

α actsas:

∂Rα f(x, y) :=

f(x, y)− f(rαx, y)xi − xj

,

where α = xi−xj and rα = (ij) switches xi and xj . Applying ⊥ to bothsides (setting y = 0) gives the map on H∗(X) ∼= Q[x]/J0.

In case α = αi is a simple root, we write ∂R(i). The operators ∂R

(i) satisfy thebraid relations ∂R

(i)∂R(i+1)∂

R(i) = ∂R

(i+1)∂R(i)∂

R(i+1), etc., so we can define ∂R

w :=∂R

(i1) · · · ∂R(i`)

for a reduced factorization w = si1 · · · si` ∈ W . Warning:unless α is a simple root, ∂R

α 6= ∂Rrα

.The left operation ∂L

α acts only on the equivariant cohomology H∗T (X).

To define it, first consider the contraction maps κα : Kα×T X → X andκT

α : (Kα×T X)T → XT , where YT := ET ×T Y for any left T -space Y .Now let ∂L

α := (κTα)∗(κT

α)∗ . For a geometric intepretation, consider a T -invariant subvariety Y ⊂ X, and left multiply by Kα to get the largerT -invariant subvariety DL

α(Y ) := KαY ⊂ X. Then we can express theDemazure operation in terms of fundamental classes as:

∂Lα : HT

∞−2`(X) −→ HT∞−2`+2(X)

[Y ]T 7−→

[DLαY ]T if dimR(DL

αY ) = dimR(Y ) + 20 otherwise .

On the polynomial ring Q[x, y], we have:

∂Lαf(x, y) := (−1)

f(x, y)− f(x, rαy)yi − yj

.

where α = yi−yj and rα = (ij) switches yi and yj . (N.B. the minus infront!) We once again have ∂L

w, and in general ∂Lα 6= ∂L

rα.

Let us emphasize that in all the above, α can be any root, not necessarilysimple. We will sometimes write ∂x

α, ∂yα instead of ∂R

α , ∂Lα , respectively.

Finally, we consider the Gysin push-forward of the projection π : X → X :

π∗ :H2`T (X) →→ H2`−2M

T (X)

f(x, y) 7−→ ∂x($)f(x, y) ,

where 2M = dimR(X) − dimR(X) , and we denote ∂x($) := ∂R

w , the rightDemazure operator of w = w$, the longest element of the maximal parabolicW$ = StabW ($) = S1 × Sn−1 . In fact, given f(x, y) ∈ H∗

T (X), we maytake its Grobner normal form f(x, y) and expand it as:

f(x, y) =∑

a=(a2,...,an)

fa(x, y) xa22 · · ·x

ann ,

where x = x1 . Then for a = (n−2, . . . , 2, 1, 0) we have:

∂x($) f(x, y) = ∂x

($) f(x, y) = fa(x, y) mod J$ .


1.13. Schubert polynomials. The basepoint pt := idT ∈ K/T = X cor-responds to ptT ⊂ XT , which is isomorphic to the diagonal diag(BT ) ⊂BT×BKBT .

The equivariant class of the basepoint, i.e., the class of the diagonal,corresponds via Poincare duality to the double Schubert polynomial:

Sw0(x, y) :=∏

i+j≤n

(xi − yj) = [pt]T ∈ H2LT (X) ,

where 2L := dimR(X) = 2(n2

).

Proof. Recall that BT ×BK BT maps to the space:

B(S1)n ×BUn

B(S1)n = [v1, . . . , vn]× [v′1, . . . , v′n] ,

where vi, v′i ∈ P∞ and [v1, . . . , vn] , [v′1, . . . , v

′n] run over orthogonal 1-dimensional

splittings of some n-dimensional space inside the Hermitean space C∞. Thenatural coordinates for H∗

T (X) = H∗(BT ×BK BT ) are the Chern classesx1, . . . , xn and y′1, . . . , y

′n, which can be defined by the loci: xi = [vi ∈ H]

and y′i = [v′i ∈ H] , where H ⊂ C∞ is a hyperplane.For i, j ∈ [1, n], consider the locus on which vi is orthogonal to v′j , the

divisor:Dij :=

vi · v′j = 0

.

The diagonal ∆BT ⊂ BT ×BK BT is the locus:

∆BT =

vi = v′i | i = 1, . . . , n

=⋂i<j

Dij ,

a transversal intersection. Demazure shows that we can express our divisorsas: [Dij ] = xi − y′j , so:

[pt]T = [∆BT ] =∏i<j

(xi − y′j) .

Substituting yi := y′n+1−i gives the desired formula.

For general w ∈W , the equivariant classes corresponding to the Schubertvarieties Xw := closure(B · wT ) ⊂ X are given by the double Schubertpolynomials Sw(x, y). These can be computed from Sw0(x, y) via the rightrecurrence:

Swsi(x, y) := ∂R(i)Sw(x, y) if `(wsi) = `(w)−1 ;

or alternatively the left recurrence:

Ssiw(x, y) := ∂L(i)Sw(x, y) if `(siw) = `(w)−1 .

Our constructions imply the non-trivial fact that these two recurrences pro-duce the same polynomial Sw for each w. This fact is equivalent to theidentity:

Sw(x, y) = (−1)`(w) Sw−1(y, x) .

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Setting y = 0 gives us the single Schubert polynomials Sw(x) := Sw(x, 0),representing the Schubert classes in the ordinary cohomology H∗(X). Thesepolynomials can be computed directly from Sw0(x) = xn−1

1 xn−22 · · ·x2

n−2xn−1

via the right recurrence: however, the left recurrence no longer makes sense.

Example: For G = SL3C , we have the equivariant cohomology ring of theflag variety X = Flag(C3) :

H∗T (X) ∼=

Q[x, y]J

∼=Q[x1, x2, x3, y1, y2, y3] x1+x2+x3 , y1+y2+y3

(x21+x1x2 + x2

2)− (x1+x2) (y1+y2+y3) + (y1y2+y1y3+y2y3)x3

1 − x21 (y1+y2+y3) + x1 (y1y2+y1y3+y2y3)− y1y2y3

,

and the double Schubert polynomials are:

Sw0 = (x1−y1)(x1−y2)(x2−y2)

Ss1s2 = (x1−y1)(x1−y2) , Ss2s1 = (x1−y1)(x2−y1)

Ss1 = x1−y1 , Ss2 = x1+x2−y1−y2 , S1 = 1 .

Setting y1 = y2 = y3 = 0 gives the ordinary cohomology ring:

H∗(X) ∼=Q[x]J0

∼=Q[x1, x2, x3](

x1+x2+x3 , x21+x1x2+x2

2 , x31

) ,

with the basis of single Schubert polynomials:

Sw0=x21x2 , Ss1s2=x2

1 , Ss2s1=x1x2 , Ss1=x1 , Ss2=x1+x2 , S1=1 .

For the projective space X = P2, we take x := x1 , and we have:

H∗T (X) ∼=

Q[x, y]J$

∼=Q[x, y1, y2, y3](

y1+y2+y3

x3 − x2 (y1+y2+y3) + x (y1y2+y1y3+y2y3)− y1y2y3

)and H∗(X) ∼= Q[x]/J$

0∼= Q[x]/(xn), with the Schubert polynomials:

Ss2s1=(x−y1)(x−y2) , Ss1= x−y1 , S1=1 .

1.14. Weyl group actions on cohomology. The non-holomorphic rightaction of W on X = K/T , namely kT · w := k

wT , induces a right action

on H∗T (X). Recall that each reflection is an orientation-reversing map.

The naive notion of a left W -action, w · kT :=wkT , is not well-defined.

However, w · Y :=wY does define an operation on T -invariant subvarieties

Y ⊂ X , so we get a left W -action on H∗T (X).

Considering XT as BT ×BK BT , we can see these two actions as thenatural W -actions on the two factors. That is, we have the natural action ofW on t∗ , written λ 7→ w(λ) or in coordinates w(xi) = xw(i) , and this induces


the usual W -action on S = Sym(t∗) . If a cohomology class is Poincare dualto a subvariety Y ⊂ X, so that:

[Y ]T = f(x, y) = f(x,w0(y′)) ∈ S ⊗SW

S ,

where y′ are the natural coordinates and y = w0(y′) , yi = y′n+1−i accordingto our convention, then:

[w ·Y ]T = f(x,w0(wy′)) = f(x,w∗(y)) , [Y ·w]T = (−1)`(w)f(w−1(x), y) ,

where w∗ := w0 w w0 is the Weyl duality, and the sign adjusts orientation.Now consider the equivariant class of a T -fixed point w = wT ∈ X .

If pt = idT is the basepoint with [pt]T = Sw0 =∏

i+j≤n(xi − yj) , thenw = w · pt = (−1)`(w) pt · w, so we have:

[w]T = Sw0(x,w∗(y)) = (−1)`(w) Sw0(w−1(x), y) mod J .

The two polynomials on the right are not equal, but equivalent modulo theideal.

Considering H∗T (X) = S$ ⊗SW S ∼= Q[x, y]/J$, we have T -fixed points

ui = uiK$ ∈ X = K/K$ , where ui = sisi−1 · · · s2s1 for i = 0, . . . , n−1 are

the minimal coset representatives of W/W$ . Their classes are:

[ui]T = Sun−1(x, u∗i (y)) =∏

j 6=n−i

(x− yj) .

Note that [ui∈X]T = π∗ [ui∈X]T .

2. Affine Schubert varieties and factorization

2.1. Minimal parabolic subgroups. The elementary unipotent subgroupassociated to a root α = ei−ej is Uα := I + xEij | x ∈ C ∼= (C,+),where Eij is an off-diagonal coordinate matrix. A minimal parabolic isthe subgroup generated by a negative elementary subgroup and the Borel:Pα = U−αB, where α is a positive root. For a simple root α = αi = ei−ei+1,we write the parabolic as P(i). We also have the compact form Kα := K∩Pα

consisting of matrices which are diagonal except for a copy of U2 in the ij-block, so that T ⊂ Kα.

We make the corresponding definitions for the loop group. If ε0 : G[[t]]→G, f(t) 7→ f(0) is the evaluation map, we define the Iwahori sugroupB := ε−1

0 (B), which is also a Borel subgroup of G((t)) thought of as a Kac-Moody group. For α = ei−ej and k ∈ Z, we have the affine root subgroupU(α,k) := I+xtkEij | x ∈ C. The positive affine roots are of the form (α, 0)and (±α, k) for α a positive root of G and k > 0. We define the minimalparabolic P(α,k) := U(−α,−k)B whenever (α, k) is positive. In particular, forthe simple roots αi := (ei−ei+1, 0) for i = 1, . . . , n−1, and α0 := (en−e1, 1),we write the minimal parabolic as P(i).

The compact form of an affine parabolic is: K(α,k) := LK ∩ P(α,k). Cru-cially, K(α,k) is once again a finite-dimensional compact Lie group. Indeed,

16 MAGYAR

K(ei−ej , k) for i < j is generated by T ⊂ LK and a copy of SU2 in theij-block, in the form: [

a −btk

bt−k a

],

where a, b ∈ C with aa + bb = 1. (This lies in SU2 for all t ∈ S1, sincet−1 = t.) Also K(ej−ei, k) = K(ei−ej ,−k), the complex conjugate of theabove.

We have K(α,0) = Kα ⊂ K ⊂ LK for all the finite roots α, and thegroups K(0),K(1), . . . ,K(n−1) corresponding to the affine simple roots areall isomorphic to each other via the diagram automorphism of Dyn. Forexample, for G = SL3C, the simple-root compact parabolics are:

K(0) =

a 0 −bt−1

0 c 0bt 0 a

, K(1) =

a −b 0b a 00 0 c

, K(2) =

c 0 00 a −b0 b a

,

where a, b, c ∈ C with (aa + bb) c = 1.

2.2. Schubert varieties in ΩK. We say the factorization v = v1 · · · vr ∈W is reduced if `(v) = `(v1)+· · ·+`(vr), and we write this as: v = v1· · ·vr.

Let v = si1 · · · si` ∈ W be a reduced factorization into affine simplereflections. The spaces we define below will be independent of the choice offactorization for v. Define the algebraic group Schubert variety:

Gv := P(i1) · · · P(ir) ⊂ G((t)) ;

and the compact group Schubert variety:

LKv := LK ∩ Gv = K(i1) · · ·K(ir) ⊂ LK .

Also define Gid := B and LKid := T . For w ∈ W , we will write Kw ⊂ Kinstead of LKw ⊂ K ⊂ LK.

The Schubert variety in the usual sense (inside the affine Grassmannian) isdenoted Grv in the algebraic category and ΩKv in the topological category:

Grv := ( Gv ·G[[t]] ) / G[[t]] ⊂ G((t))/G[[t]] ∼= Gr

∼= ΩKv := (LKv ·K) / K ⊂ LK/K ∼= ΩK .

Put another way, let ν : LK → ΩK, f(t) 7→ f(t)f(1)−1 be the base-pointnormalizing map: then ΩKv = ν(LKv).

We may also obtain the Schubert varieties in Gr ∼= ΩK as the B-orbitclosures: Grv = closure (B · v ⊂ Gr), where

v ∈ G((t)) is a representative of

v. However, this definition has no compact counterpart: the Schubert cellB · v is non-compact, so it cannot be the orbit of any compact group.

Note that ΩKv = ΩKu whenever vW = uW ; thus the distinct Schubertvarieties in ΩK are indexed by the right cosets W/W = tλWλ∈Q∨ . An-other useful set of coset representatives is given by the elements of minimal


length mλ := min(tλW ) : we call these the right-minimal elements of W .We denote:

ΩKλ := ΩKtλ = ΩKmλ and LKλ := LKmλ .

We have a homeomorphism:8

ΩKλ := (LKλ ·K)/K ∼= LKλ/ T .

Clearly, LKv and ΩKv are finite-dimensional, and one can show thatGrλ

∼= ΩKλ is a rational complex algebraic variety with dimR = 2 `(mλ).Indeed, one can construct ΩK explicitly as a CW-complex with one even-dimensional cell B · tλ ∼= C`(mλ) corresponding to each Schubert variety, sothat the fundamental cycles form a Q-linear basis of the loop-group homol-ogy:

H∗(ΩK) =⊕

λ∈Q∨

Q [ΩKλ] .

2.3. Schubert varieties in ΩKad. We will also need the Schubert subva-rieties of ΩKad, indexed by W/W ∼= P∨ ∼= Σ × Q∨ , where σ ∈ Σ are theelements of length zero. We say v = v1 · · · v` ∈ W is a reduced factoriza-tion into extended simple roots if we take vj ∈ s0, s1, . . . , sn−1 ∪ Σ , andthe product is reduced (using the length function on W ). For each σ ∈ Σand its representative

σ ∈ LK, we let Kσ :=

σT = T

σ ⊂ LK. Now for any

v ∈ W , we can define LKv, ΩKv, etc., word-for-word as before, except thatnow ΩKv ⊂ ΩKad, etc. For v ∈ W , we consider ΩKv ⊂ ΩK ⊂ ΩKad.

To any λ ∈ P∨ we associate λ ∈ Q∨ as follows. Take mλ = σλ mλ withσλ ∈ Σ and mλ ∈ W . Since mλ is again right-minimal, we have mλ = m

bλfor some λ. Further, let ηλ = σλ(0) be the minuscule fundamental weight(or the zero weight) corresponding to σλ. Then we have:

ΩKλ = ΩKmλ = Kσλ· ΩKbmλ = σλ · ΩKbλ ⊂ ΩKad .

2.4. Bott-Samelson varieties. Now suppose v = v1v2 · · · vr is any fac-torization of v ∈ W (or v ∈ W ), not necessarily reduced. We define theBott-Samelson variety as the quotient:

ΩKv1,...,vr = LKv1

T× · · ·

T×LKvr/ T :=

LKv1× · · · × LKvr

T × · · · × T

where LKv1× · · ·×LKvr has the following right action of T r:

(x1, . . . , xr) · (t1, . . . , tr) := (x1t1 , t−11 x2t2, · · · , t−1

r−1xrtr) .

This is homeomorphic to the usual algebraic definition of the Bott-Samelsonvariety (cf. Demazure). Indeed ΩKv1,...,vr is a complex algebraic variety withdimR = 2`(v1) + · · ·+ 2`(vr).

8Indeed, the obvious map LKλ/T → (LKλ ·K)/K is a continuous bijection betweencompact Hausdorff spaces, and so is a homeomorphism.

18 MAGYAR

Now let v = v1 · · · vr with v right-minimal (which implies that vr isalso right-minimal). Then we have ΩKv

∼= LKv/ T , and we can define thecontraction map:

cont : ΩKv1,...,vr → ΩKv

(x1, . . . , xr) 7→ x1· · ·xr .

This map is continuous and surjective, and it is birational, meaning thatcont−1(U) ∼= U for an open neighborhood U of almost any point of ΩKv.Therefore the induced homology map cont∗ : H∗(ΩKv1,...,vr) → H∗(ΩK)takes the fundamental class of the Bott-Samelson variety to that of theSchubert variety: cont∗[ΩKv1,...,vr ] = [ΩKv].

2.5. Factorization. Recall that mλ := min(tλW ) ; that mλλ∈Q∨ , whereQ∨ is the coroot lattice, are coset representatives for W/W ; and that theSchubert varieties of ΩK are ΩKλ := ΩKmλ = ΩKtλ for λ ∈ Q∨.

Furthermore, mλλ∈P∨ , where P∨ is the coweight lattice, are coset rep-resentatives for W/W , where W is the extended affine Weyl group. Forλ ∈ P∨, we decompose mλ = σλmλ with `(σλ) = 0 and mλ ∈ W , and wedefine λ ∈ Q∨ by m

bλ = min(mλW ).

Theorem A: Suppose we have coweights λ, µ, ν ∈ P∨ with µ ∈ P∨− anti-

dominant, such that mλ tµ = mν ∈ W and `(mλ)+ `(tµ) = `(mν). Then theSchubert homology classes in the Pontryagin ring H∗(ΩK) satisfy:

[ΩKbλ] · [ΩKbµ] = [ΩKbν ] .Proof. For any µ ∈ P∨

− (meaning −µ ∈ P∨+ is dominant), we have mµ = tµ

and `(wtµW ) ≤ `(tµW ) for all w ∈ W . Thus K · LKµ · K = LKµ · K,meaning that LKµ ·K has a left K-action.

Given mλ tµ = mν as in the Theorem, we can write the correspondingBott-Samelson variety as:

ΩKλ,µ := LKλ

T×LKµ/ T ∼= (LKλ ·K)

K× (LKµ ·K)/K .

Now we can define a homeomorphism by the two inverse maps:

ΩKλ,µ∼←→ ΩKλ × ΩKµ

|| ||

(LKλ ·K)K×(LKµ ·K)/K (LKλ ·K)/K × (LKµ ·K)/K

( f1(t) , f2(t) ) 7−→ ( f1(t) f1(1)−1 , f1(1) f2(t) )

( f1(t) , f2(t) ) ←−| ( f1(t) , f2(t) ) .


Furthermore, we have the commutative triangle:

ΩKλ,µ∼−→ ΩKλ×ΩKµ

cont ↓ mult

ΩKν ,,

where mult is loop group multiplication and cont is the birational contractionmap. Taking the induced maps on the fundamental classes in the homologyH∗(ΩK) gives:

[ΩKλ] · [ΩKµ] = mult∗[ΩKλ×ΩKµ] = cont∗[ΩKλ,µ] = [ΩKν ] .

Now we have [ΩKλ] = [σλ] · [ΩKbλ] , where [σλ] is a one-point homologyclass, and:

[σλ] · [ΩKbλ] · [σµ] · [ΩKbµ] = [σν ] · [ΩKbν ] .But Pontryagin multiplication is commutative and σλσµ = σν , so we cancancel the one-point homology classes from the above equation.

We make some comments on the proof. The key to defining the home-omorphism ΩKλ,µ

∼→ ΩKλ × ΩKµ is to pick out the K-factor of a loopf1(t) ∈ LKλ · K : it is the basepoint f1(1). This allows us to disentanglethe K-twisted product. Let us see why this proof cannot be adapted tothe algebraic category (and so does not prove the analogous results aboutDemazure modules). Algebraically, we can again write the Bott-Samelsonvariety as:

Grλ,µ := (Gλ ·G[[t]])G[[t]]× (Gµ ·G[[t]]) / G[[t]] ,

which is homeomorphic to ΩKλ,µ . But now there is no algebraic way topick out the G[[t]]-factor of f1(t) ∈ Gλ ·G[[t]] , so we cannot disentangle theG[[t]]-twisted product. In fact, Grλ,µ and Grλ × Grµ have non-isomorphiccomplex structures which are connected by deforming the complex structureon a fixed underlying topological space, as in the recent work of Beilinson-Drinfeld (see [G]).

3. Affine Schubert polynomials

3.1. Homology symmetric algebra. Let X = K/K$ ∼= Pn−1, where$ := $∨

1 , the first fundamental coweight. We have the singular homology:

H∗(X) := H∗(Pn−1, Q) = Q1⊕Qh1 ⊕ · · · ⊕Qhn−1 ,

where 1 is the 0-dimensional homology class of a point and hi is the 2i-dimensional homology class of a projective i-plane Pi ⊂ Pn−1. For ourpurposes, H∗(X) is a vector space with no natural multiplication.

Recall that H∗(Xm) ∼= H∗(X)⊗m. Consider the embedding pt : Xm ∼→pt ×Xm ⊂ Xm+1 : by abuse of notation, pt means either the basepoint ofX or the inclusion map. Define the symmetric power SmH∗(X) as the quo-tient of H∗(X)⊗m by the commutativity relations, and define the symmetric

20 MAGYAR

algebra as the direct limit

SH∗(X) = lim−→

SmH∗(X)

under the above embeddings, which induce:

pt∗ : SmH∗(X) → Sm+1H∗(X) , y1 · y2 · · · ym 7→ 1 · y1 · y2 · · · ym .

In particular, the unit is 1 = 1 · 1 = · · · and any y ∈ H∗(X) is identifiedwith y = 1 · y = 1 · 1 · y = · · · . In our case, we clearly have SH∗(X) ∼=Q[h1, · · · , hn−1] , a polynomial ring.

3.2. Bott’s isomorphism Ψ. Bott defines9 the mapping:

Ψ : K → ΩKk 7→ k t$ k−1 t−$ .

This descends to a map Ψ : X = K/K$ → ΩK, and it induces a Q-linearmap Ψ∗ : H∗(X) → H∗(ΩK) and a ring homomorphism Ψ∗∗ : SH∗(X) →H∗(ΩK).

Theorem: The Bott map Ψ∗∗ is an isomorphism from the polynomial ringQ[h1, . . . , hn−1] = SH∗(Pn−1) to the Pontryagin ring H∗(ΩK). This iso-morphism preserves the dimension grading, where dim(hi) = 2i.

In the sections that follow, we will paraphrase Bott’s proof of this theorem(at least of the surjectivity part). Our aim is to explicitly describe for eachλ ∈ Q∨ the unique polynomial Sλ ∈ SH∗(X) with Ψ∗∗(Sλ) = [ΩKλ].

Note: For G of other types, the map Ψ is defined in terms of a weight $with the property that for each simple root α1, α2, . . . of G, there existssome w ∈W such that 〈αi, w($)〉 = 1. (There is always some fundamentalcoweight $ = $∨

i with this property.) Then we take X = K/K$ withH∗(X) ∼= Q1⊕Qh1 ⊕ · · · ⊕Qhr, where the basis elements hj correspond tothe cosets in the parabolic Bruhat order W/W$.

The map Ψ∗∗ is always surjective, as will be clear from our argumentsbelow. However, Ψ∗∗ might not be injective, so that Sλ = Ψ−1

∗∗ [ΩKλ] isno longer a unique polynomial. Rather, H∗(ΩK) will be isomorphic toa quotient of the polynomial ring SH∗(X) = Q[h1, . . . , hr] by a certainpolynomial ideal I, and Sλ will be well-defined modulo I. The algebraicvariety Spec H∗(ΩK) ⊂ Cr defined by the ideal I is described in Peterson’snotes, and is often called the Peterson variety.

3.3. Bott’s map Φ. We will find it convenient to rephrase Bott’s isomor-phism in terms of the Z-shifted map:

Φ :K/K$→ ΩKad

k 7→ k t$ k−1 .

9Our mappings Ψ, Φ are in Bott’s notation gs, fs respectively.


We have induced maps Φ∗ : H∗(X) → H∗(ΩKad) and Φm∗ : H∗(Xm) →

H∗(ΩKad). However, since Φ∗(1) = t$ 6= 1 ∈ ΩKad, the maps Φm∗ do not

intertwine with the inclusions SmH∗(X) → Sm+1H∗(X), and do not inducea map Φ∗∗ : SH∗(X)→ H∗(ΩKad).

Instead, we consider the normalized maps

Φm(x1, . . . , xm) := Φ(x1) · · ·Φ(xm) · t−m$ ,

which do induce a map Φ∗∗ : SH∗(X) → H∗(ΩK). Because of the commu-tativity of the Pontryagin product, we clearly have Φ∗∗ = Ψ∗∗ .

Lemma: Let λ ∈ P∨ correspond to λ ∈ Q∨, and let [Y ] ∈ H∗(Xm). Then:

Φm∗ [Y ] = [ΩKλ] ∈ H∗(ΩKad) =⇒ Ψ∗∗[Y ] = [ΩKbλ] ∈ H∗(ΩK) .

Proof: Let [tλ] denote the zero-dimensional homology class of the point tλ ∈ΩKad. Let η be the minimal dominant weight with η = m$ mod Q∨: sincetη and tm$ lie in the same component of ΩKad, we have [tη] = [tm$]. Indeed,Im(Φm) lies completely in the (m$)-component of ΩKad, and Φm

∗ [Y ] =[ΩKλ], so we have λ = m$ mod Q∨ also, and ΩKλ = ΩKbλ · tη. Then:

Φm∗ [Y ] = [ΩKλ] = [ΩKbλ] · [tη] = [ΩKbλ] · [tm$] .

Hence Ψ∗∗[Y ] = Φ∗∗[Y ] = Φm∗ [Y ] = [ΩKλ] · [t−m$] = [ΩKbλ], proving the

Lemma.

3.4. Schubert varieties pulled back to Xm. Given v = mλ ∈ W and areduced factorization into simple reflections, we can group the factorizationinto elements of W separated by the affine simple reflection s0 :

v = w1 s0 w2 s0 · · · wr s0 ,

where wj ∈W . Recall that:

s0 = tθ∨/2 rθ t−θ∨/2 = t$ rθ t−$ ,

where θ = x1 − xn is the highest root, since 〈θ , θ∨/ 2〉 = 〈θ, $〉 = 1 . Ex-pressing this in terms of root subgroups K(0) ⊂ LK and Kθ ⊂ K ⊂ LK, wehave:

K(0) = K(−θ,1) = t$Kθt−$ .

We may write:

t−$ = t−π(e1) = tπ(e2+e3+···+en)

= tu1($) tu2($) · · · tun−1($)

= u1t$u−1

1 u2t$u−1

2 · · · un−1t$u−1

n−1 ,

where:ui = sisi−1 · · · s1 = (i+1, i, . . . , 2, 1) ∈W = Sn

is an (i+1)-cycle with ui(1) = i+1.

22 MAGYAR

Now we can express the Schubert class ΩKv ⊂ ΩK in terms of the mapΦ as follows:

ΩKv = (Kw1t$Kθt

−$) · · · (Kwr t$ Kθ t−$)/T

= (Kw1 t$ Kθ u1 t$u1 u2t$u−1

2 · · · un−1t$u−1

n−1) · · ·= k1 t$ k−1

1 · (k1k′1 u1) t$ (k1k

′1 u1)−1 · (k1k

′1u2) t$ (k1k

′1u2)−1 · · ·

= Φ(k1) · Φ(k1k′1u1) · · ·Φ(k1k

′1ur) · Φ(k1k

′1k2) · Φ(k1k

′1k2k

′2u1) · · · ,

where kj runs over Kwj and k′j runs over Kθ for j = 1, . . . , r. Hence wedefine the submanifold Y ⊂ Xm = (K/K$)m, where m = nr, as:

Y :=

(k(1), k′(1)u1, . . . , k′(1)un−1,

k(2), k′(2)u1, . . . , k′(2)un−1,...

k(r), k′(r)u1, . . . , k′(r)un−1)

∣∣∣∣∣∣∣∣∣k(j) := k1k

′1 · · · kj−1k

′j−1kj , k′(j) := k(j)k

′j

k1∈Kw1 , . . . , kr∈Kwr , k′1, . . . , k′r∈Kθ

,

so that Φm(Y ) = Φm(Y ) = ΩKv = ΩKλ, and Ψ∗∗[Y ] = [ΩKλ]. Thus,we have reduced our problem to evaluating the fundamental class [Y ] as apolynomial in Q[h1, . . . , hn−1].

We remark that Y is an embedding of a Bott-Samelson variety:

Y ∼= Kw1

T×Kθ

T× · · ·

T×Kwr

T×Kθ/T ∼= ΩKw1,s0,...,wr,s0 .

Indeed, we can express the Bott-Samelson variety of ΩK purely in termsKα ⊂ K by virtue of the isomorphism of (T×T )-spaces K(0)

∼= Kθ.

3.5. Demazure operations on multi-projective space. We considerthe multi-projective space Xm, where X = K/K$ ∼= Pn−1 and m = nr .For a root α of W , we recall the geometric Demazure operation Dα = DL

α

which takes a T -invariant submanifold Z ⊂ Xm to the new T -invariantsubmanifold:

Dα(Z) := Kα · Z ⊂ Xm .

(Because we will use only left operations, we omit the superscript L.)For a finite Weyl group element w = si1 · · · si` ∈ W , we can define

Dw := D(i1) · · ·D(i`), which is independent of the reduced decomposition.(Once more, Dα 6= Drα .) Also we recall the base point pt = idK$ ∈ Xand the T -fixed points ui = ui · pt ∈ X for i = 0, . . . , n−1 . Also, we abusenotation so that w(k) ∈ W k can mean either a T -fixed point of Xk or aninclusion map Xm ∼→ w(k) ×Xm ⊂ Xk+m , and similarly with pt .

Now let us use these operations to construct the manifold Y ⊂ Xm asso-ciated to the reduced factorization v = w1 s0 · · · wr s0 in the previoussection. Denote u(n−1) := (u1, . . . , un−1) ∈Wn−1 . Then we clearly have:

Y = Dw1 ptDθ u(n−1) Dw2 ptDθ u(n−1) · · ·Dwr ptDθ(u(n−1)) .

We proceed to interpret in polynomial terms each of these operations.


Consider the equivariant cohomology H∗T (Xm) := H∗(ET×T Xm). By

the same argument as for the case m = 1, we have:

ETT× Xm ∼=

m factors︷︸︸︷BT ×

BK· · · ×

BKBT ×

BKBT .

Taking H∗T (pt) ∼= S ∼= Q[y1, . . . , yn]/(y1+ · · ·+yn) , we get:

H∗T (Xm) ∼= S ⊗

SWS ⊗

SW· · · ⊗

SWS

∼=Q[x(1), x(2), . . . , x(m), y](

y1+ · · ·+ yn = 0h(x(1)) = · · · = h(x(m)) = h(y) , ∀h ∈ SW

) ,

where x(k) = (x(k)1 , . . . , x

(k)n ) and y = (y1, . . . , yn), and the dimension grading

becomes dim(x(k)j ) = dim(yj) = 2. We also have:

H∗T (Xm) ∼= S$ ⊗

SWS$ ⊗

SW· · · ⊗

SWS$ ⊗

SWS

∼=Q[x1, x2, . . . , xm, y](

y1+ · · ·+ yn∑nj=0 (−1)j x j

i en−j(y) , i = 1, . . . ,m

) ,

where xk := x(k)1 .

The (left) geometric Demazure operation Dα induces via Poincare dualitythe cohomology operation:

∂yα : H2`

T (Xm) → H2`−2T (Xm)

f(x, y) 7→ (−1)f(x, y)− f(x, rαy)

yi − yj,

where α = yi−yj ; and we again have ∂yw for w ∈W .

Recall the top-degree Schubert polynomial of H∗T (X) = S$ ⊗SW S , rep-

resenting the basepoint of X :

[pt∈X]T = Sun−1(x, y1, . . . , yn) =n−1∏j=1

(x− yj) ,

as well as the other T -fixed points:

[ui ∈ X]T = Sun−1(x, u∗i (y)) =∏

j 6=i+1

(x− yj) .

The embedding u(n−1) : Xk 7→ Xk+n−1 induces the map:

[u(n−1)]T f(x1, . . . , xk, y) :=Sun−1(x1, u

∗1(y)) · · ·Sun−1(xn−1, u

∗n−1(y)) f(xn, . . . , xk+n−1, y) .

We also have the fiber mapping ⊥ : H∗T (X)→ H∗(X) , f(x, y) 7→ f(x, 0) .

24 MAGYAR

Now we write:

[Y ] = ⊥ ∂yw1

[pt]T ∂y−θ [u(n−1)]T ∂y

w2[pt]T ∂−θ [u(n−1)]T · · · ∂y

wr[pt]T ∂y

−θ [u(n−1)]T .

This is a cohomology class:

[Y ] ∈ H2m(n−1)−2`(Xm) ∼=[

Q[x1, . . . , xm](xn

1 , . . . , xnm)

]dim=2m(n−1)−2`

where ` = `(mλ) := r+`(w1)+· · ·+`(wr) and dim(xj) = 2 . This is Poincaredual to the fundmental class:

[Y ] ∈ H2`(Xm) ⊂ SpanQ(1, h1, . . . , hn−1)⊗m .

Symmetrizing induces the affine Schubert polynomial:

Sλ = Sym [Y ] ∈ SymmSpanQ(1, h1, . . . , hn−1) = Q[h1, . . . , hn−1]deg≤m ,

This has dim(Sλ) = 2` , where the dimension grading is defined by dim(hi) =2i ; and deg(Sλ) ≤ m = nr , where deg means the usual grading deg(hi) = 1 .

3.6. Combinatorics of Schubert classes. In this section we collect theprevious arguments into a self-contained combinatorial recipe to constructour affine Schubert polynomials:

Theorem B: For any λ ∈ Q∨, the polynomial

Sλ ∈ Q[h1, . . . , hn−1] = SH∗(X)

defined by the steps (1)–(6) below satisfies Ψ∗∗(Sλ) = [ΩKλ] ∈ H∗(ΩK).

(1) We have the affine Weyl group of periodic permutations of Z:

W = π ∈ SZ | π(i + n) = π(i) + n ,

which is the semi-direct product of the finite Weyl group W = Sn ,and the root lattice:

Q∨ =

tλ∣∣∣∣ λ = (λ1, . . . , λn) ∈ Zn

λ1 + · · ·+ λn = 0

,

where tλ(i) := i + nλ(i mod n) . This is a Coxeter group generated bythe simple reflections si = (i, i+1) ∈ W for i = 1, . . . , n−1 , and bys0 = rθt

θ , where rθ = (1, n) and θ = (1, 0, . . . , 0,−1) , the highestcoroot.

The cosets W/W have minimal representatives mλ = min(tλW )for λ ∈ Q∨. We take a reduced factorization mλ = w1 s0 · · · wr s0

into finite Weyl group elements w1, . . . , wr ∈ W separated by theaffine simple reflection s0 ∈ W .


(2) We let m = rn, and we work with m + n variables x = (x1, . . . , xm)and y = (y1, . . . , yn) . We define the ring:

Q[x, y]J$

:=Q[x1, . . . , xm, y1, . . . , yn](

y1+ · · ·+ yn = 0∑nj=0 (−1)n−j x j

k en−j(y) , k = 1, . . . ,m

) ,

where ei(y) denotes an elementary symmetric polynomial. The givenpolynomials are a Grobner basis of the ideal under the term ordery1 < · · · < yn < x1 < · · · < xm .

(3) For a root α = yi − yj , we have the divided difference operator:

∂yαf(x, y) :=

f(x, y)− f(x, rα(y))yi − yj

,

where rα switches the variables yi and yj . Here α need not be a sim-ple root: for example, it may be the longest root θ = y1 − yn . Forw = si1 · · · sip ∈ W a reduced factorization into simple reflectionsof W , we define ∂y

w := ∂yαi1· · · ∂y

αip.

(4) For i = 1, . . . , n , we have the twisted point-class polynomial:

P(k)i :=

n∏j=1

j 6=i

(xk − yj) ,

and for an interval of n−1 integers

[a+1, a+n−1] = a+1, a+2 . . . , a+n−1 ,

we define:

P[a+1,a+n−1] := P(a+1)n−1 P

(a+2)n−2 · · · P

(a+n−1)1 .

(5) To the factorization mλ = w1 s0 · · · wr s0 we associate a poly-nomial S(x, y) ∈ Q[x, y]/J$ by repeatedly applying the divided dif-ference operators and multiplying by point-class polynomials:

S(x, y) := ± ∂yw1

P(1)n ∂y

θ P[2, n] ∂yw2

P(n+1)n ∂y

θ P[n+2, 2n] · · ·

· · · ∂ywr

P(rn−n+1)n ∂y

θ P[rn−n+2, rn] ,

where the sign ± = (−1)`(w1)+···+`(wr) takes care of orientations.We specialize our polynomial to y = 0 :

S(x) := S(x, 0) ∈ Q[x]J$

0

:=Q[x1, . . . , xm](xn

1 , . . . , xnm)

.

Note that deg(S) = m(n−1)− `(mλ) .

26 MAGYAR

(6) We consider Q[x]/J$0 as a Q-vector space with a basis of monomials

xi11 · · · xim

m with 0≤ ik≤n−1 . Now we take the Q-linear mapping:

Sym : Q[x]/J0 −→ Q[h1, . . . , hn−1]

xi11 · · · xim

m 7−→ hn−i1−1 · · ·hn−im−1 ,

where h0 := 1. The mapping xik 7→ xi corresponds to symmetrizing

the tensor product, and the mapping xi 7→ hn−i−1 undoes Poincareduality, turning a cohomology class into a homology class.

Finally, we define:

Sλ(h) := Sym S(x) .

This is a polynomial with degree ≤ rn and with dimR Sλ(h) =`(mλ), where dimR(hi) := 2i . It is independent of the choice ofreduced factorization for mλ.

3.7. Affine Schubert polynomials for SL3 . Let us first apply the aboverecipe in the case n = 3. We can deduce from Theorem A that all Schubertclasses are products of three “prime” classes:

Sρ , S−$1 , S−$2 .

Here we have ρ := $1+$2 , and we write $i instead of $∨i for legibility.

Also, for λ ∈ P∨, we denote Sλ := Sbλ where λ ∈ Q∨ is defined by mλ =

σλmbλ with `(σλ) = 0 . We have:

W = 〈s0, s1, s2, σ〉 where σsiσ−1 = s(i+1 mod 3) .

To simplify notation further, we will write xk instead of xk. We have:

P(k)i := (xk−y1) (xk−y2) (xk−y3) / (xk−yi) .

We compute as follows:

• λ = ρ = $1+$2 , mλ = s0 , r = 1 , m = 3 .

S(x, y) := P(1)3 ∂y

13 P(2)2 P

(3)1

= (x1 − y1)(x1 − y2)(x3 − y2)(x2 − y3)(x2 − y1)

S(x) = x21x

22x3 , S(h) = h1 .

• λ = −$1 , mλ = σ2s2s0 , r = 1 , m = 3 .

S(x, y) := ∂32P(1)3 ∂y

13 P(2)2 P

(3)1

= (x1 − y2)(x2 − y1)(x1x2 − x1x3 + x2x3 − x2y2 − x2y3 + y2y3)

S(x) = x21x

22 − x2

1x2x3 + x2x22x3

S(h) = h2 − h21 + h2

1 = h2 .


• λ = −$2 , mλ = σs1s0 , r = 1 , m = 3 .

S(x, y) := ∂21P(1)3 ∂y

13 P(2)2 P

(3)1

= −(x2 − x3)(x2 − y3)(x1 − y1)(x1 − y2)

S(x) = x21x2x3 − x2

1x22 , S(h) = h2

1 − h2 .

• λ = −2$2 , mλ = σ2s0s2s1s0 , r = 2 , m = 6 .

S(x, y) := P(1)3 ∂y

13 P(2)2 P

(3)1 ∂32∂21P

(4)3 ∂y

13 P(5)2 P

(6)1

S(x) = −x21x

22x3(x3 − x4)(x5 − x6)(x4 − x5)

S(h) = (h21 − h2)2 , consistent with Theorem A.

3.8. Affine Schubert polynomials for SL4 . For n = 4, we list the λ ∈P∨ corresponding to the “prime” classes Sbλ ∈ Q[h1, h2, h3].

• λ = −$1 , mλ = σ3s2s3s0 , Sbλ = h3 .

• λ = −$2 , mλ = σ2s0s3s1s0 , Sλ = h22 − h1h3 .

• λ = −$3 , mλ = σs2s1s0 , Sλ = h31 − 2h1h2 + h3 .

• λ = ρ = $1+$2+$3 , mλ = σ2s2s1s3s0 , Sλ = h2(h21 − h2) .

• mλ = σ2s1s3s0 , Sλ = h1h2 − h3 .

• mλ = σ2s3s0 , Sλ = h2 .

• mλ = σ2s1s0 , Sλ = h21 − h2 .

• mλ = σ2s0 , Sλ = h1 .

Notice the bit of mysterious extra factorization for Sρ .

3.9. Calculations for higher SLn . For general n, the “prime” Schubertclasses include S−$i , the classes corresponding to the fundamental anti-dominant translations t−$i = m−$i . (We again identify $i with $∨

i , etc.)We can decompose this as:

t−$i = (sisi−1 · · · s2s1σ)n−i = wiw0σ−i ,

where wi = maxW$i , the longest element of a maximal parabolic, and σ isthe automorphism of the fundamental alcove with:

σsiσ−1 = si+1 , where sn := s0 .

Now, λ = −$i 6∈ Q∨, so Sλ means Sbλ , where λ ∈ Q∨ is defined by mbλ =

σit−$i . Indeed, Shimozono has calculated that, letting k := min(i, n−i),

28 MAGYAR

we have:

λ = kαi +k−1∑j=1

(k−j)(αi−j + αi+j) .

For example for n = 6, we have:

−$1 = α1 −$2 = α1 + 2α2 + α3

−$3 = α1 + 2α2 + 3α3 + 2α4 + α5 .−$5 = α5 −$4 = α3 + 2α4 + α5

The other prime classes are Sλ for λ ∈ Q∨ with (mλ)−1 lying inside thefundamental box, as described in the Appendix. The fundamental box hasmaximal alcove

mρ = tρw0 ∈ W ,

where ρ =∑n−1

i=1 $i , which is also the half-sum of the positive coroots. Thiselement is an involution:

(mρ)−1 = w0tρ = t−w0(ρ)w0 = mρ .

We can decompose this as mρ = ι mρ, where mρ ∈ W and:

ι =

1 for n oddσn/2 for n even .

The alcoves in the fundamental box are the right weak order interval [1, m]R ,and the corresponding prime classes are:

[1, m−1]L = [1, ι m ι]L =

w ∈ W | ι m ι = u w for some u

.

4. Appendix: Weak order on an affine Weyl group

This note addresses the following question concerning optimal factoriza-tion in an affine Weyl group.

4.1. Question. Given an affine Weyl group element x ∈ W , minimal inits coset xW with respect to the finite Weyl group W , find the maximaldominant coweight λ such that there exists a factorization x = y · t−λ with`(x) = `(y) + `(t−λ), where y ∈ W , the extended affine Weyl group, andt−λ ∈ W is an anti-dominant translation.

4.2. Answer. Let $∨1 , . . . , $∨

n be the fundamental coweights of W . For avector γ = c1$

∨1 + · · · cn$∨

n with ci ∈ R, define its floor as the integralcoweight:

bγc := bc1c$∨1 + · · ·+ bcnc$∨

n ,

where bcc denotes the integer floor of a real number.Choose a tiny positive vector ε := ε1$

∨1 + · · · + εn$∨

n , where εi > 0 aresufficiently small positive real numbers. Then the dominant coweight askedfor is: λ = bx−1(ε)c, which is independent of the choice of ε.


4.3. Examples.

a. For x = wtµ, where w ∈W and µ is a dominant coweight, this gives:

λ = µ−∑

i∈D(w−1)

$∨i ,

where D(w−1) := i ∈ [1, n] | w−1(αi) < 0, the descent set of w−1.b. Let W be of type A

(1)2 and x = s1s0. We have s1($∨

1 ) = −$∨1 +$∨

2 ,s2($∨

2 ) = $∨1 −$∨

2 , and si($∨j ) = $∨

j for 1 ≤ i 6= j ≤ n. Also, s0 =tθ∨rθ, where θ∨ = $∨

1 + $∨2 and rθ($∨

1 ) = −$∨2 , rθ($∨

2 ) = −$∨1 .

Then ε = ε1$∨1 + ε2$

∨2 , and x−1(ε) = (1−ε1−ε2)$∨

1 + (1+ε2)$∨2 , so

λ = 0$∨1 + 1 $∨

2 = $∨2 .

c. Let W be of type C(1)2 with Dynkin diagram 0⇒1⇐ 2, and x =

s0s1s0. We have s1($∨1 ) = −$∨

1 + 2$∨2 , s2($∨

2 ) = $∨1 −$∨

2 , andsi($∨

j ) = $∨j for 1 ≤ i 6= j ≤ n. Also, s0 = tθ∨rθ, where θ∨ = $∨

1

and rθ($∨1 ) = −$∨

1 , rθ($∨2 ) = −$∨

1 + $∨2 . Then ε = ε1$

∨1 + ε2$

∨2 ,

and x−1(ε) = ε1$∨1 + (2−2ε1−ε2)$∨

2 , so λ = 0$∨1 + 1 $∨

2 = $∨2 .

4.4. Root systems and alcoves. We proceed to prove the above answer,as well as record some general facts for reference(cf. [H], [S]).

Let Rn and (Rn)∗ be dual vector spaces. A positive definite inner productallows one to identify Rn and (Rn)∗, but we will use only the natural pairing〈 , 〉 between (Rn)∗ and Rn. We define a root system by a set ∆ ⊂ (Rn)∗

of root vectors and a set ∆∨ ⊂ Rn of coroot vectors. We have the bases ofsimple roots α1, . . . , αn ∈ (Rn)∗ and simple coroots α∨1 , . . . α∨n ∈ Rn, togetherwith their dual bases, the fundamental weights $1, . . . , $n ∈ (Rn)∗ andfundamental coweights $∨

1 , . . . , $∨n ∈ Rn. For α ∈ ∆, we have the reflection

rα : Rn → Rn , µ 7→ µ−〈α, µ〉α∨. The simple reflections s1, . . . , sn generatethe finite Coxeter group W .

For a vector λ ∈ Rn we have the translation tλ : Rn → Rn , µ 7→ µ + λ,so that w tλ w−1 = tw(λ) for w ∈ W . We define the affine Weyl groupW = W ∝ tQ∨ and the extended affine Weyl group W = W ∝ tP∨ , whereQ∨ = Zα∨1 ⊕ · · · ⊕ Zα∨n is the coroot lattice and P∨ = Z$∨

1 ⊕ · · · ⊕ Z$∨n is

the coweight lattice. As a Coxeter group, W has generators s0, s1, . . . , sn,where s0 = tθ∨ rθ = rθt−θ∨ .

The affine reflections rα,k ∈ W correspond to the hyperplanes

Hα,k = µ | 〈α, µ〉 = k

for α ∈ ∆+ and k ∈ Z. The (closures of) connected components of Rn \Hα,k are called alcoves: they are simplicies provided ∆ is an irreducibleroot system. We define the fundamental alcove

A0 := µ | 0 ≤ 〈α, µ〉 ≤ 1 for all α ∈ ∆+= µ | 〈α1, µ〉, . . . , 〈αn, µ〉 ≥ 0 , 〈θ, µ〉 ≤ 1 .

30 MAGYAR

This is a fundamental domain of W acting on Rn, and each alcove A corre-sponds to a unique w ∈ W with A = wA0. Henceforth we identify w ∈ Wwith its alcove wA0.

Since the neighbors of A0 are siA0, any two neighboring alcoves A,A′ areof the form A = wA0 and A′ = wsiA0 for some i = 0, . . . , n; and we labelthe common facet of A and A′ with i. The length function on W (or on theset of alcoves) is defined as: `(w) = # hyperplanes separating w from 1 .

The extended W is the symmetry group of the affine hyperplane ar-rangement Hα,k, or of the set of alcoves, and the quotient W/W isisomorphic to the symmetry group of A0 itself: σA0 = A0 for each min-imal coset representative σ ∈ W/W . That is, W = Σ ∝ W , whereΣ := σ ∈ W | σ(A0) = A0 . Every element x ∈ W can be uniquelyfactored as:

x = ∧xσ = σ x∧

where σ ∈ Σ and ∧x, x∧ ∈ W . We have xA0 = ∧xA0, and we define `(x) :=`(∧x) = `(x∧). In particular, for each λ ∈ P∨ we have tλ = ∧tλ σ with∧tλ ∈ W and some σ ∈ W/W .

We also have the fundamental chamber C0 = µ | 〈α1, µ〉, . . . , 〈αn, µ〉 ≥0, and a translation tλ lies in C0 exactly when λ is dominant. Furthermore,for λ a dominant coweight, wtλ lies in the chamber wC0, and t−λ lies in−C0 = w0C0.

Lusztig has given the following length formula for wtλ ∈ W , where w ∈Wand λ ∈ P∨:

`(wtλ) =∑

α∈∆+

|〈λ, α〉|+ `(w) .

4.5. Strong and weak orders. There are three important orders on W

and W . The strong or Bruhat orderB≤ has covering relations x

B≤ rα,kx,

where rα,k is any reflection with `(rα,kx) = `(x) + 1. We have xB≤ y iff

x−1B≤ y−1.

In this note, we will mainly consider the weak orders: the left weak orderL≤ has covers x

L≤ six, where si (i = 0, . . . , n) is a simple reflection with

`(six) = `(x) + 1. The analogous right weak orderR≤ has covers x

R≤ xsi.

Since xL≤ y iff x−1

R≤ y−1, the left and right orders are combinatorially

equivalent. However, the right order is easier to picture in terms of alcoves:

the covering relation xR≤ xsi steps from an alcove A = xA0 to its neighbor

A′ = xsiA0 through a facet labelled i, provided the facet separates A′ fromthe fundamental A0.


These orders become very simple on dominant translations tλ. That is,let λ, µ be dominant coweights. Then we have:

tλB≤ tµ ⇐⇒ µ−λ ∈ Q∨

+ := Nα∨1 ⊕ · · · ⊕ Nα∨ndef⇐⇒ λ

Q≤ µ ,

∧tλR≤ ∧tµ ⇐⇒ µ−λ ∈ P∨

+ := N$∨1 ⊕ · · · ⊕ N$∨

ndef⇐⇒ λ

P≤ µ .

In the second line we must use ∧t instead of t, since we can only have tµR≤ tλ

when tµ ≡ tλ mod W , meaning µ− λ ∈ Q∨.When restricted to an orbit W · tλ for λ ∈ P∨

+ , the affine Bruhat or-der becomes the finite Bruhat order on the parabolic quotient W/Wλ. IfWλ = 1, we have for w, y ∈W :

wtλB≤ ytλ ⇐⇒ w

B≤ y =⇒ w(λ)

Q≥ y(λ) .

Warning: The converse of the last implication does not hold.10 To charac-terize the weak order on W , define the inversion set:

Inv(w) := ∆+ ∩ w−1∆− = α(1), . . . , α(n) ,

where α(k) := sinsin−1 · · · sik+1(αik) . Now we have: y

L≤ w iff Inv(y) ⊂

Inv(w) , and yR≤ w iff Inv(y−1) ⊂ Inv(w−1) . To define the strong and

left weak orders on W/Wλ, we replace the cosets wWλ by their shortest (orlongest) representatives and take the orders induced from W .

The orders can be adapted to parabolic quotients of the affine Weyl group.Consider W/W = tλWλ∈Q∨ , the right cosets of the affine Weyl group bythe finite Weyl group. Pictorially, each coset corresponds to a translatetλP0 of the Weyl polytope P0 :=

⋃w∈W wA0 by an element of the coroot

lattice. Let min(xW ), max(xW ) denote the minimal and maximal element

in the coset. We define: `(xW ) = `(min(xW )), and xWL≤ sixW whenever

`(sixW ) = `(xW ) + 1. In fact,

xWL≤ yW ⇐⇒ min(xW )

L≤ min(yW ) ⇐⇒ max(xW )

L≤ max(yW ) .

The left cosets Wx ∈ W\W can best be pictured by their minimal rep-resentatives, the alcoves inside the fundamental chamber: A ⊂ C0 withA = min(Wx) = tλw, where λ ∈ Q∨

+ and w ∈W with w minimal in the coset

w StabW (λ). We define WxR≤Wy so that: xW

L≤ yW iff Wx−1

R≤Wy−1.

10Example: W = Sn+1 , λ = ρ = (n+1, n, · · · , 2, 1) ∈ P∨+ ⊂ Rn+1. We have:

wB

≤ y ⇐⇒ w(1), . . . , w(i) ≤ y(1), . . . , y(i) for i = 1, . . . , n ,

w(ρ)Q

≥ y(ρ) ⇐⇒ w(1)+ · · ·+w(i) ≤ y(1)+ · · ·+y(i) for i = 1, . . . , n .

For w = 2314 , y = 4123 , the second condition holds but not the first.

32 MAGYAR

The weak orders extend in an obvious way to W/W and W\W . TheBruhat order on W/W can be defined similarly, and I believe the minimalcovers of xW = vtλW for λ ∈ P∨

+ and v ∈W are:

vtλWB≤

rαv tλW if `(rαv) = `(v) + 1rαv tλ−α∨i

W if `(rαv) = `(v)− 1 .

I do not know a good criterion for when vtλB≤ wtµ.

4.6. Factorization and weak order. A reduced factorization is an expres-sion x = y · z, where x = yz ∈ W and `(x) = `(y) + `(z). This is closelyrelated to the weak order: for x, y, z ∈ W ,

x = y · z ⇐⇒ xL≤ z ⇐⇒ x

R≤ y .

For x, y, z ∈ W , this becomes:

x = y · z ⇐⇒ x∧L≤ z∧ ⇐⇒ ∧x

R≤ ∧y ,

since if x = y · z with y = ∧yσ, z = ∧zσ′, then x = ∧xσσ′ and ∧x =∧y · (σ ∧z σ−1) with σ ∧z σ−1 ∈ W . The converse means: if ∧x

R≤ ∧y, then for

any σ, σ′′ ∈ Σ, there exists z ∈ W such that ∧xσ′′ = ∧yσ · z. Similarly for x∧.Now we can rephrase the Question. Given x ∈ W , minimal in its coset

xW , we want the maximal λ ∈ P∨+ such that for some y ∈ W , we have:

x = y · t−λ, meaning x−1 = tλ · y−1, meaning ∧tλR≤ x−1.

4.7. Question reformulated: Given x−1 ∈ W , minimal in its coset Wx−1,

find theP≤-maximal coweight λ ∈ P∨

+ such that ∧tλR≤ x−1.

This has an obvious pictorial answer. The group tP∨ acts on Rn with fun-damental domain equal to the parallelopiped spanned by the fundamentalcoweights:

B0 := µ | 0 ≤ 〈αi, µ〉 ≤ 1 for i = 1, . . . , nWe call B0 the fundamental box, and its translates Bλ := tλB0 cover Rn =⋃

λ∈P∨ Bλ. Each box is a union of N alcoves, where N = |W | / [W : W ] =|W | / [P∨ : Q∨]. Furthermore, the fundamamental chamber is the union ofthe dominant boxes: C0 =

⋃λ∈P∨+

Bλ.

Now we have: ∧tλR≤ y iff yA0 ⊂ Bλ. (The ⇒ is easy. The ⇐ is because

. . . ) Hence, given y = x−1 ∈ W we have:

maxµ | ∧tµR≤ y = λ where yA0 ⊂ Bλ .

Now, if we take any point ε = ε1$∨1 + · · · + εn$∨

n in the interior of A0, wehave:

yA0 ⊂ Bλ ⇐⇒ y(ε) ∈ Bλ ⇐⇒ ∀ i : 〈αi, λ〉 < 〈αi, y(ε)〉 < 〈αi, λ〉+1 .


Therefore, the answer to our question is exactly:

λ := b〈α1, y(ε)〉c$∨1 + · · ·+ b〈αn, y(ε)〉c$∨

n ,

which was to be shown.

References

[B] R. Bott, The Space of loops on a Lie group, Michigan Math Journal (1958).[BS] R. Bott and H. Samelson, Application of the theory of Morse to symmetric spaces,

American Journal of Mathematics 80 (1958), 964–1029.[G] D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby

cycles, Inventiones Mathematicae 144 (2001), 253–280.[H] Hiller, The Geometry of Coxeter Groups (book).[L] Thomas Lam, Schubert polynomials for the affine Grassmannian, preprint

arxiv.org/math.CO/0603125[P] Dale Peterson, MIT Course 18.735, Spring 1997, notes by Monica Nevins and Con-

stanze Rietsch.[S] Jian-yi Shi, Alcoves corresponding to an affine Weyl group, J. London Math. Soc.

(Ser. 2) 35 (1987), pp. 43-74.

Peter [email protected] of Math, Wells HallMichigan State UniversityEast Lansing, MI 48823

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